Our proof is a generalization
of the proof of [1]}.
Let \(S\)  be a logarithmic scheme.
Following the discussion after Definition REF  we may consider \(\mathcal {J}^{trop}_{g,n}\)  as a category over \(\textbf {ShpMon}^{op}\) . Then \(\mathcal {J}^{trop}_{g,n}(\Gamma (S,\overline{M}_S))\)  consists of pairs \((\Gamma ,D)\)  with \(\Gamma \)  metrized by the monoid \(\Gamma (S,\overline{M}_S)\) .
We can therefore associate to \(S\) 
a natural map \(\mathcal {J}^{trop}_{g,n}(\Gamma (S,\overline{M}_S))\rightarrow \widetilde{\mathcal {J}}^{trop}_{g,n}(S)\)  that is given by associating to \((\Gamma ,D)\)  in \(\mathcal {J}^{trop}_{g,n}(\Gamma (S,\overline{M}_S))\)  the collection \(\big \lbrace (\Gamma _s/\overline{M}_{S,s}, D_s)\big \rbrace _s\)  of contractions of \((\Gamma ,D)\)  induced by \(\Gamma (S,\overline{M}_S)\rightarrow \overline{M}_{S,s}\)  (for each geometric points \(s\)  of \(\underline{S}\) ) together with the obvious morphisms \(\pi _f:(\Gamma _s)\rightarrow (\Gamma _t)\)  induced by the generalization maps \(\overline{M}_{S,s}\rightarrow \overline{M}_{S,t}\)  (for every étale specialization \(f\colon t\rightsquigarrow s\)  of geometric points of \(\underline{S}\) ).
This map is natural in \(S\)  and therefore, since \(a^\ast \mathcal {J}^{trop}_{g,n}\)  is a stack, it induces a morphism \(a^\ast \mathcal {J}^{trop}_{g,n}\rightarrow \widetilde{\mathcal {J}}^{trop}_{g,n}\) .
We argue that this is an isomorphism, i.e. that every \(\big \lbrace (\Gamma _s/\overline{M}_{S,s},D_s)\big \rbrace _s \in \widetilde{\mathcal {J}}^{trop}_{g,n}(S)\)  lifts uniquely, up to unique isomorphism, to an object in \(a^\ast \mathcal {J}^{trop}_{g,n}\) . Since both \(a^\ast \mathcal {J}^{trop}_{g,n}\)  and \(\widetilde{\mathcal {J}}^{trop}_{g,n}\)  are stacks, this is an étale local assertion on \(S\) . So we may assume that there is a geometric point \(s_0\)  of \(S\)  and a chart \(\overline{M}_{S,s_0}\xrightarrow{} \Gamma (S,\overline{M}_{S})\) .
Using this homomorphism,
we obtain a natural lift \((\Gamma ^{\prime },D^{\prime })\in \mathcal {J}^{trop}_{g,n}(\Gamma (S,\overline{M}_S))\)  of \(\big \lbrace (\Gamma _s/\overline{M}_{S,s},D_s)\big \rbrace _s\) .
This lift is unique since the composition
\(\mathcal {J}^{trop}_{g,n}(\Gamma (S,\overline{M}_S))\rightarrow \widetilde{\mathcal {J}}^{trop}_{g,n}(S)\rightarrow \widetilde{\mathcal {J}}^{trop}_{g,n}(s_0)=\mathcal {J}^{trop}_{g,n}(\overline{M}_{S,s_0})\) 
is an isomorphism. Let \(\big \lbrace (\Gamma ^{\prime \prime }_s/\overline{M}_{S,s},D_s^{\prime \prime })\big \rbrace _s\)  be the image of \((\Gamma ^{\prime },D^{\prime })\)  in \(\widetilde{\mathcal {J}}^{trop}_{g,n}(S)\) . Then, by [1]}, the isomorphism \((\Gamma ^{\prime \prime }_{s_0}/\overline{M}_{S,s_0},D_{s_0}^{\prime \prime })\simeq (\Gamma _{s_0}/\overline{M}_{S,s_0},D_{s_0})\)  extends to an étale neighborhood of \(s_0\)  and so the lift \((\Gamma ^{\prime },D^{\prime })\)  is unique up to unique isomorphism.
The second assertion, namely that \(\widetilde{\Phi }^{trop}\cong a^*\Phi ^{trop}\) , follows from the compatibility of this isomorphism with stabilization.
